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The Art of Fugue, BWV1080

Introduction

The Art of Fugue is universally upheld as a major intellectual tour de force of Western civilisation; one of the great wonders of musical art. Above all it summarizes the entire known potential of counterpoint.

Standard dictionaries define counterpoint thus:

n. the art of combining melodies. adj. contrapunt'al [Fr. contrepoint and It. contrappunto—L. contra, against, punctum, a point. points or notes placed against those of the melody.

Etymology is well and good. But the form is only understood when its principles are fully realised as they are within this masterpiece. No-one understood the workings of counterpoint more fully than Bach and in his final years the composer, then Cantor at St Thomas’s, Leipzig, was more and more preoccupied with contrapuntal music. He completed the ‘Goldberg Variations’ (1742), the Musical Offering (1747, see above) and variations on ‘Vom Himmel hoch’ (1747).

By 1748 he was ready for a great, final summation; a work with every manner of contrapunct and canon based on one great theme. The following year work commenced and, despite the ravages of a fatal eye disease, this final profound undertaking was almost completed in the year of his death (1750). It proved a monumental revelation, an unfinished series of contrapuntal variations imbued with unfailing variety and limitless imagination.

While creating The Art of Fugue, the 65-year-old Bach embodied within the work unparalleled splendour and poetry. But his overriding aim was purely to exhibit the comprehensive possibilities of a single, simple ‘subject’ with various types of fugal and canonic writing. The debilitating final disease prevented its completion. But during his final months, work on publishing The Art of Fugue had already begun.

A complete version in Bach’s autograph predates the published one and the formal copper engraving was partially supervised by him. However, it could hardly be said to bear his imprimatur for at some point members of the family began passing pages to the ‘unknown’ engraver who continued working from the manuscript with no thought or understanding of the music or its true sequence. Bach’s son Carl Philipp Emanuel eventually took charge of the publication and it appeared posthumously in autumn 1751. The results were messy and bewildering and performers looked at the work in utter confusion. The Art of Fugue was regarded as a labyrinthine exercise; a drily academic tangle of uncommon severity.

This widely held view meant that Bach’s towering masterpiece suffered undeserved neglect and obscurity for much of its history. Carl Czerny and Philipp Spitta regarded it as a keyboard work. But difficulties arose from the incomplete form. Bach had not finished the final fugue, superscribed ‘Fuga a soggeti’ in the printed version. And aspects of the original printing were the subject of unending speculation.

The opening four fugues pose no problems. Indeed there is no great stumbling block up to Contrapunctus 11. Most academics detect Bach’s influence in the ordering of the pieces to this point though several ponder over the use of three (not four) ‘stretto’ counter fugues; Contrapuncti 5-7.

Still greater difficulties arise with the remaining unnumbered items, so quixotically arranged in the first print. Until the present century many would-be performers remained doubly flummoxed as the score bore no directions about the work’s instrumentation. Credit for its twentieth-century revival must go to the young Swiss student Wolfgang Graeser (1906–28). He painstakingly unravelled the tangle and in 1924 prepared a running order with the canons and fugues set for various groups of instruments. The Graeser version was first heard in 1927 at a concert in the Leipzig church of St Thomas. Karl Straube directed.

Interest was enormous. Musicians everywhere took note and the academic fraternity considered the event a watershed in our understanding of eighteenth-century music. Beyond the hub of Bach’s continent, The Art of Fugue was now reconsidered with equal zeal, thought of less as the last disordered, creative gasp of a dying man and more the comprehensive, consummate summation of Bach’s immeasurable genius.

News of this event spread through Europe like wildfire. It crossed the Atlantic and within two years Stokowski presented Graeser’s orchestral version at Mrs Coolidge’s Chamber Music Festival in the Library of Congress, Washington. By 1930 and 1931 New York audiences were witness to performances by the Juilliard Graduate School under director Albert Stoessel.

In fact Graeser had simply split the work into two parts, each one beginning with limited forces (string quartet or harpsichord). From this base the instrumentation was progressively enlarged, first to chamber ensemble proportions and then to full orchestral dimensions. In the first eleven fugues Graeser adopted the order of the original; his first half comprised the four simple fugues, the three inverted fugues and the four double/triple fugues. His second part had the canons, mirror fugues and the quadruple fugue.

Graeser’s arrangement was soon set aside in favour of more economical instrumentations. Weighty orchestral incarnations all but vanished. By this time the work was most usually heard with chamber orchestras or from string quartets. Bach’s four-part writing led to still more experimentation with a variety of groupings. Organists turned to The Art of Fugue and the work was re-examined yet again.

In 1932 Tovey endorsed the nineteenth-century belief that the work had been intended for the keyboard. He published an open-score edition as well as one for keyboard. Like other performers and academicians he also produced a ‘complete’ version of the final, four-part fugue which breaks off after measure 239. Whether Bach intended his variations for the organ or harpsichord, for chamber group or orchestra, remains unclear to this day.

Gustav Leonhardt contends that a mere glance at the compass of the alto voice in the first twelve fugues will reveal how none of Bach’s ensemble groups may properly be used in performing the work. ‘Every instrumentation must resort to a completely anachronistic group of instruments,’ he says. And he adds, ‘no single voice has a specific instrumental character. This … may account for the greater variety of instrumental attempts.’ The Dutch harpsichordist and scholar explains that Bach never used the soprano clef for flute, oboe or violin. But the clefs he does specify were widely accepted for classical polyphony and equally for keyboard instruments.

As we have seen, the nature of instrumentation is not documented. Moreover it appears to have been submitted as an abstract counterpoint, independent of any particular instrumental setting. Such observations have led to a paradoxical theory that Bach was wholly unconcerned about the eventual performance. Which raises the question—was The Art of Fugue merely set down as an elaborate intellectual exercise?

German scholar Friedrich Blume reasoned that Bach saw his work as an esoteric activity, a disinterested transmission of purely abstract theory: ‘Bach wanted to continue a tradition of consummate contrapuntal skill … inherited from the (Roman) school of Palestrina’s period by way of Sweelinck, Theile, Werckmeister and Vitali.’

C P E Bach himself thought the work’s greatest value was as a teaching aid. He declared, ‘Every student of the art … cannot fail to learn from it how to compose a good fugue and will therefore need no oral teacher, who often charges dearly enough.’ Schweitzer took a similar view in his J S Bach, le musicien-poète (1905).

By 1756 The Art of Fugue had sold only thirty copies and 120 years later it was similarly overlooked but for the sporadic publication of a few keyboard editions. After the attentions of Graeser and Tovey in the late ’20s and early ’30s, a number of keyboard artists revived the work. Each one argued persuasively that The Art of Fugue was most properly suited to his/her own instrument.

Leonhardt put the case for the harpsichord and Helmut Walcha claimed it as an organ work. Today’s music dictionaries usually espouse a diplomatic, less self-serving viewpoint, merely noting: ‘a keyboard performance would seem most obvious.’ The fact that it was published in score is immaterial. F W Marpurg’s added preface (1752) explains that this was to facilitate reading. Charles Rosen says eighteen of the most complex contrapuntal works do not fall by chance within the compass of two hands. He also comments, ‘The Art of Fugue was meant to be studied by playing it, to have its marvels seen, heard, and felt under one’s fingers—it must indeed, be played many times before its deceptive lucidity can be penetrated. There are almost no dramatic effects; the most fantastic modulations take place discreetly, and the sequences are continually varied with a delicacy unparalleled in Baroque music.’

Bach began his monumental task with four fugues, two presenting the theme, the others running in contrary motion. Added to this were counterfugues where the first statement is inverted and recombined with its original form. Then there were double and triple fugues, four canons and two pairs of mirror fugues. Karl Geiringer notes: ‘To make the mirror reflection doubly realistic, the treble of the first fugue becomes the bass of the second, the alto changes into a tenor, the tenor into an alto and the bass into a treble.’ The inversus appears like the rectus standing on its head.

In fact the order of pieces on the autograph differs from that established from the printed version. The autograph is demonstrably in Bach’s hand while details of the printed version are, even now, not finally authenticated. Numbering of the first (scrupulously engraved) pieces may be based on a second ‘lost’ autograph, changed from the original layout by Bach himself. Ordering of the mirror fugues remains the subject of prolonged and inconclusive debate.

Recordings

'The remarkable Tatiana Nikolayeva … uncovers Bach's revelations as if they were her own' (The Independent)'Absolutely superb. Certainly I can think of no finer recording of this endlessly fascinating masterpiece, whatever the medium. Highly recommended' (C ...» More

Angela Hewitt’s much-awaited recording of Bach’s ultimate masterpiece, The Art of Fugue, is destined to be the crowning achievement of her Bach cycle for Hyperion—a revelatory recording and performing project which has taken her all over the world ...» More

Details

Contrapuncti 1, 2, 3 and 4 are the four simple fugues as detailed below. The first of four canonical fugues; ‘alla Ottava’ takes its place between these introductory contrapuncti and three counter fugues viz. Contrapunctus 5, Contrapunctus ‘in Stylo Francese’ and Contrapunctus ‘per Augmentationem et Diminutionem’. The second canonical fugue; ‘alla Duodecima in Contrapunto alla Quinta’ serves to heighten the distinction between Bach’s three counterfugues (as above) and the four fugues with several themes (Contrapunctus 8, 9 [‘alla Duodecima’], 10 [‘alla Decima’] and 11). Two remaining canonical fugues lie between the fugues with several themes and the two mirror fugues. Canon ‘per Augmentationem in Contrario Motu’, with its forthright variation of the basic form of the principal theme, relates strikingly to Contrapunctus 11. Then Canon ‘alla Decima, Contrapuncto alla Terza’ with the syncopated inversion heralds the first of the two mirror fugues. These are numbered Contrapunctus 13 (Rectus and Inversus) and Contrapunctus 12 (Rectus and Inversus) and are so ordered through a process of simple logic. Together they lead to Contrapunctus 14, the closing B-A-C-H fugue. The twenty tracks can be further detailed as follows:

Contrapunctus 1: The first of four simple fugues. Here it is played straight and serves to introduce the basic shape and character of the theme.

Contrapunctus 4: Again Bach employs the same inverted form of his theme. This is a simple four-part fugue with a chromatically moody counter-subject. There is an oblique descending-third reference to a hymn utilised in the cantatas: viz. ‘Wer weiss, wie nahe mer mein Ende’ (‘Who knows how near my end might be’) by the Countess of Schwarzburg-Rudolstadt (1637–1707).

Canon (in Hypodiapason) alla Ottava: One of four canonical fugues. In two parts; the main theme is at first inverted, then played straight with substantial rhythmic changes and embellishments. It develops as a simple canon at the octave and the lower part follows the upper at four bars’ distance.

Contrapunctus 6: In four parts (‘in Stylo Francese’). Once more the main theme is varied with passing notes. The response is inverted and diminished while variously-spaced stretti are straight, inverted, normal, diminished, double and triple. The theme takes on four seperate guises.

Contrapunctus 7: In four parts (per Augmentationem et Diminutionem). Additional to variants of the basic theme heard so far. The exposition with its inverted response passes from bass to soprano and the augumentations are enriched by the four versions of the theme from Contrapunctus 6. Stretti occur at a variety of intervals.

Canon alla Duodecima in Contrapunto alla Quinta: In two parts. Another (straight) version of the main theme with major changes of rhythm and embellishments. Within the first section the canon is in the upper duodecima. For the second section parts are interchanged with the canon now in the double sub-octave. The upper part follows the lower at a distance of eight bars to the middle of the fugue; then the upper part leads with the lower trailing by eight bars (at the octave).

Contrapunctus 8: A three-part triple fugue and one of the most outwardly attractive items of the entire work. Two new themes are heard in straight configuration and reappear in contrapunctus 11. Later these themes combine with the variation of the principal theme in its inverted from. Now the main theme is inverted and rythmically diversified.

Contrapunctus 9 (alla Duodecima): Following the exposition of this new theme, the main augmented theme reappears seven times at differing intervals as a fixed melodic accompaniment. On each occasion the new theme and principle theme combine in two different intervals.

Contrapunctus 10 (alla Decima): Another new theme is introduced. Its shift from straight to inverted form occurs after the exposition. Bar 23 brings a second exposition. This time passing notes serve to vary the inverted main theme. There is also a vital new counter-subject. The two themes are combined after bar 44 while passages of thirds and sixths further enliven both themes.

Contrapunctus 11: A four-part triple fugue. It begins with a variation of the rythmically altered principal theme (cf. the inverted form at Contrapunctus 8). The themes combine and recombine in both straight and inverted form; eight themes in all. This complex equation includes an ascending/descending chromatic line regarded as the fourth theme.

Canon (in Hypodiatessaron) per Augmentationem in Contrario Motu: A daring variation on the basic outline of the theme. Its lower part follows the upper in augmented inversion at four bars’ distance. The latter half has this same canon. But now, from mid-way its parts are interchanged.

Canon alla decima 'Contrapunto alla terza': Bach’s technique repeats that on. Again there are two parts with the main theme changed. The fugue starts out as a canon at the tenth (upper decima). It is changed mid-way, becoming a canon at the sub-octave. The crux of this fugue is its simple, syncopated inversion.

Contrapunctus 13 (Rectus): A mirror fugue in three parts. The simple, basic form of the theme is constantly changed, embellished and inverted. Its non-thematic bass remains exempt. Both mirror fugues are extended for two harpsichords where required.

Contrapunctus 13 (Inversus): A complete (mirror) inversion of the preceding Rectus. The middle section of the former becomes the upper part. By the same token the lower part takes the middle, and the upper part is bass of the Inversus.

Contrapunctus 12 (Inversus): The entire four-part fugue is mirrored with its bass becoming the soprano and so forth. Now the total inversion has its own autonomous validity. In our present century Fricker welds the mirror and its image (12 Studies for Piano) while Bartók’s ‘Chromatic Inversion’ (Mikrokosmos VI) allows for simultaneous or successive images on two pianos.

Contrapunctus 14: In this final triple fugue Bach chose to include the letters of his own name (B = B flat and H = B natural in German nomenclature). In this way he set his personal seal on the work as a whole. The first deliberate, ricercare-like theme reads the same whether backward or forwards. It is heard in stretto and in inversion. The second theme, principally in quavers, eventually combines with the first. The third (final) theme turns to the B-A-C-H motif, presented in minims. After a brief passage of counterpoint the print ends while the autograph continues for another seven bars. Hearing the BACH theme, the instant of unison with all three themes, then abrupt and sudden silence, gives the work a final shattering impact.

Bach’s crowning achievement is concluded. But Tatiana Nikolayeva should surely have the final word in this note. Between recording ‘takes’ at Rosslyn Hill Chapel, Hampstead, she recalled a concert in Kharkov, Southern Russia. When this work was over, a concert-goer told her The Art of Fugue had brought him to a closer understanding of life’s tragedies. ‘There is a great secrecy about this music,’ she concludes. ‘We shall never know Bach’s innermost thoughts—all we can do is try to discover the meaning.’

Contrapunctus 1: For a work that is so complicated, Bach could not have chosen a simpler beginning. There are no fugal games in the first contrapunctus, only entries of the subject (as indeed is the case with the first four contrapuncti). Ties and syncopated rhythms play a big role in the accompanying material, but there is no formal counter-subject. We have one overlapping entry at bar 32 (1'15—note how Bach always includes the tail of the subject, those final four descending notes, something he never neglects to do in the whole Art of Fugue); one false entry in the alto in bar 48 (1'51); a pedal point at bar 63 leading to a quasi-cadenza-like passage with dramatic pauses (2'24); and the final entry of the subject in the tenor over an extended pedal point 2'53). These last five bars were an afterthought and didn’t exist in the original version. Much of the episodic material has the same up-and-down movement (for instance in bars 17–23 (0'40–0'54) and 44–49 (1'42–1'53)), giving it a beautiful sense of line, and allowing for satisfying breathing spaces between entries. The mood is lyrical and expressive.

Contrapunctus 2: By the simple addition of some dotted notes to the tail of the subject, Bach completely alters the character of his theme. By continuing this dotted rhythm throughout the entire contrapunctus, he gives us a piece that really has some swing! At the beginning he slurs each group of four notes in this dotted rhythm. To me this has always meant that they should not be interpreted as notes inégales (in the French manner) but played exactly as written. I don’t think it means you have to play them completely legato. Especially when there is a tie to the next group (as in bar 5 in the left hand—0'08), a slight lift before the syncopated note makes the whole thing a bit jazzier. Not surprisingly, this dotted tail forms the basis of all the accompanying material. In bar 44 (1'16) we have an ascent into the subject in the soprano, presented in F major, changing the colour in a very attractive way. The entrance in the bass in bar 61 (1'46), having found itself back in the key of D minor, needs special emphasis. The tenor in bar 69 (2'00) presents its subject off the beat, gradually finding its way back home. It is interesting that in the original Berlin autograph, the piece ends in the dominant at bar 78 (2'16). It is then followed by what we know today as Contrapunctus 5. As Tovey points out, this must mean that Bach envisaged at least some of the work being performed in sequence.

Contrapunctus 3: The third contrapunctus is a beautifully expressive piece. Quietly assured, chromatic and very vocal in character, it was originally placed second in the cycle. By changing it to third position and inserting Contrapunctus 2, Bach gives us the chance to alternate moods more effectively. It presents the subject only in its inversion, now accompanied by a chromatic counter-subject, crawling up and down. Unusually he inserts a two-bar episode (bars 13–14; 0'30–0'34) before the last entrance of the subject in the exposition. The subsequent episode (bars 19–22; 0'44–0'53) has the upper two voices exchanging motifs while the bass provides accompaniment. A variation of this same episode will be repeated in bars 39–42 (1'31–1'41), providing a nice symmetry. In bar 23 (0'53) we have the first appearance of an ornamented version of the subject using syncopation and passing notes. Its third appearance in F major (bar 35 in the tenor—1'21) adds some beautiful warmth to the harmony. From then on, the ornamented version is presented alongside its original. The fugue takes six bars to wind down after the final entrance of the subject. In its original version there were only four bars at this point, the piece ending in bar 70 (2'44). Bach really knew how to improve things even more.

Contrapunctus 4: Contrapunctus 4 is one of the best fugues in the whole cycle. It didn’t exist in the early version, so it must have been written in Bach’s final years. The subject is again inverted, this time with its initial entry starting on the dominant A instead of the tonic D. The episodes are mainly based on two motifs: that four-note tail is everywhere—right way up and upside down; and along with it, often combined together (such as in the episodes which begin at bar 53 (1'38) and bar 103 (3'10)), is what Tovey calls a ‘cuckoo-like figure of a descending third’, which Bach inverts in bar 69 (2'07), turning it into a leap of a sixth. Special attention must be given to an enhanced form of the subject beginning with the entry in bar 61 (1'53). The fifth note is pushed up a step, bringing about a series of modulations that peaks at bar 79 with the G natural in the soprano (2'26). Pop music picked up that trick long ago. Knowing he couldn’t outdo that for a while, Bach then gives us an episode that lasts an amazing 26 bars (bars 81–107; 2'29–3'17). He has one more trick up his sleeve in this tremendous fugue, even though he still doesn’t allow any serious games. Beginning at bar 107 (3'17), the subject is presented in syncopated doubling—first in the lower voices in thirds, and then answered by the upper ones in sixths. One more enhanced subject in the tenor voice is followed by the original in D minor in the alto, bringing this fugue to a perfect close. The tempo should be more flowing than the preceding fugue, not solely as a contrast but because a lot of the material (especially the episode beginning at bar 53 (1'38), for instance) demands it.

Contrapunctus 5: Now the fun begins. Contrapunctus 5 uses a variant of the subject (which includes passing notes but no syncopation), presenting it in both the original and inverted forms. The tempo needs to be flowing but not hurried in order to sing all the parts well. At bar 33 (1'05) Bach introduces his first stretto of The Art of Fugue, and he begins with a close one: in contrary motion, the soprano enters one beat after the bass. You can see why he chose the variant with the passing notes for this initial stretto. The part-writing is so smooth and perfect—anything but stilted. Two more pairs of stretti enter, bringing us to the first of two ‘mirror’ episodes (bar 53; 1'46). Here the initial part of the subject is used in a four-part canon, at an even closer distance than the first stretto since the opening note is shortened in length. It is very ingenious and very much in the style of old-fashioned vocal counterpoint. Another stretto, this time not in contrary motion, has its tail lead us downwards to a close in the dominant. Then at bar 65 (2'10) we have the mirror image of the episode at bar 53. For the final five bars of this fugue (2'51) Bach expands the texture to six voices and gives us simultaneous entries of the subject in mirror image over a tonic pedal and tierce de Picardie (i.e. a cadence in the tonic major).

Contrapunctus 6 ‘in stylo Francese’: Contrapunctus 6 is a hard nut to crack. Not only do you have to deal with a very dense stretto fugue with inversions and diminutions piled on top of each other, but you also shouldn’t play exactly what is written in the score. Bach gives this piece the subtitle ‘in stylo Francese’, meaning that the rhythmic alterations so loved by the French should be adopted. In the opening four bars, for example, I play the first entry in the bass as written (I tried double-dotting it but for me this didn’t work—it led to all sorts of complications that didn’t make sense), but I double-dot the diminished version in the soprano, and then in the alto. This means that the two Gs in the right hand at the end of the third bar do not fall together (0'11). Now apply that to all 79 bars (over six pages) and you’ll see what I mean! Some ornamentation is also necessary, as well as notes inégales and proper articulation (tiny lifts in the right places). The demisemiquaver flourish that first appears at the end of bar 7 (0'25) is typical of the French manner and needs to be shortened even further. Once you have mastered the note values and sorted out all the entries, you still need to make sense of the fugue as a whole and find its road plan. The amazing thing is that the first closure in the piece comes only at the very end, and not before. If the whole fugue were played in the same dynamic range it would be boring beyond belief. Finding points where the sound needn’t be forced (such as bars 20 (1'14) and 62 (3'57), to take just two examples) is crucial for the overall impact. Contrapunctus 6 is terrifying in its magnificence, and is perhaps the one point in The Art of Fugue where I wish my piano were instead an organ.

Contrapunctus 7 ‘per augmentationem et diminutionem’: For this next contrapunctus, Bach has a grand plan. Besides presenting the subject in its regular and diminished forms, he also uses it for the first time in augmentation, taking eight bars to get through it. The augmented subject begins in bar 5 in the bass (0'19), and then works its way upwards to the tenor (bar 23; 1'37), the alto (bar 35; 2'29), and finally to the soprano (bar 50; 3'30), where it sings out triumphantly. Above, below and in between those entries, the other three parts are engaged in a constant exchange and stretto of the subject in its other forms, including a double diminution (first heard in the soprano in the second half of bar 7—0'30). There is only one three-bar episode in the entire 61-bar fugue (bars 32–34; 2'14). I choose to keep the opening very quiet for some time, because it is a long way to the end and the music needs transparency. The entrance in the tonic key in the tenor halfway through bar 36 (2'35) is the first big climax and brings the only point, two bars later (2'41), where we have all three forms of the subject at once (augmented, original, diminished). Bach gives us a deceptive cadence in bar 60 (4'10), delaying the only close in the piece by another two bars.

Contrapunctus 8: In Contrapunctus 8 we take a big step into the realm of double and triple fugues that will occupy Bach for the next four numbers. If you look at this great triple fugue without knowing anything about it, you might not at first see the motto theme anywhere in sight. Instead, at the beginning, you see a totally new subject—strutting firmly from one D to another an octave below. This is the first contrapunctus to have only three voices rather than four, but that doesn’t mean it has less strength. On the contrary, it has tremendous scope and grandeur. After 39 bars that deal only with this subject, in creeps a second one, underneath the first (1'13). It has more rhythm than melody, with its insistent repeated notes. At the end of this the bass comments with a new figure (bar 42; 1'18) that becomes significant material (also in its inversion) from now on. This second section, in which the first two subjects are constantly combined, is brought to a half-close in bar 93 (2'53) with something of a flourish. Then what happens? Introduced by that important motif in the bass, a third subject appears in the alto (bar 94; 2'55), which, in the manner of Brahms, has silences on every first beat. You could be forgiven for not immediately recognizing the inverted motto theme.

I often like to show students in masterclasses how Bach chooses contrasting material for his subjects and countersubjects so that they easily stand out from each other (and this is why we must also choose a different articulation for each). This is a perfect example. One subject has leaps; another has repeated notes but remains tight; the third is expressive and legato. They all cleverly begin at a different part of the bar.

After the exposition of this third subject in all the voices, and more combinations of the first two, we finally come to the first triple counterpoint—all three subjects together (bar 147; 4'34). You might expect something grand for that. But no—they creep in very quietly, almost unnoticed. The first one starts off in A minor and ends up in F major. Before the end of this stunning fugue, Bach gives us four different positions of these combined subjects, gradually building up, with the help of another flourish and a pedal point, to the final triumphant entry of the motto theme in the bass.

Contrapunctus 9 ‘alla duodecima’: If virtuosity only means playing in a fast tempo then, up until now, it hasn’t been required. That changes with Contrapunctus 9. Nothing much, however, is gained by playing it presto. In fact Bach had second thoughts about this piece. In the original Berlin autograph, it was placed fifth and the note values were shorter (i.e. instead of quavers, there were semiquavers and the time signature was 4/4 instead of alla breve). In fact that is the case with all the double and triple fugues of the cycle (Nos 8–11). The last Prelude of the Well-Tempered Clavier (B minor, Book II) underwent the same change. No doubt these alterations were made to steer people away from playing them too fast. As Tovey rightly points out, bars 20–21 (0'22) in Contrapunctus 9 would sound ridiculous in too quick a tempo. The running first subject is combined with the motto theme beginning in bar 35 (0'39). With all the scampering around, there is no need for the motto theme to appear in anything but its original, simple version. In bars 85–88 we have two false starts in the soprano and alto (1'37–1'42) before it finally takes off in the bass. This contrapunctus is written in double-counterpoint at the interval of a twelfth, which means the two subjects can swap positions (the upper voice becoming the lower voice) and produce a whole new set of harmonies.

Contrapunctus 10 ‘alla decima’: In its original form, the double fugue of Contrapunctus 10 began in what is now bar 23 (1'02—with the bare subject, not including the bass). When you know only the later version, this is hard to imagine. The opening page is, to quote Tovey, ‘one of the profoundest and most beautiful [Bach] ever wrote’. The rests that we encountered in the variation of the motto theme in Contrapunctus 8 (and that we find again in Contrapunctus 11) return in this initial subject, which, along with two sighing figures, contains a row of gently undulating quavers. It is an expressive gesture, not to be hurried. And already on that first page Bach hints at the main reason for writing a piece in double-counterpoint at the interval of the tenth, which is the doubling in thirds and sixths that this makes possible (0'44). After two of the four voices enter, he then gives us the other two in stretto and in contrary motion. When he does get around to the motto theme in bar 23 (1'02), the voices are spaced to allow maximum clarity. In bar 44 (1'57) Bach combines both subjects together in the inner voices. After some very expressive episodic material, the motto theme is presented in sixths in bar 75 (3'15), with the first subject supporting it in the bass. Then at bar 85 (3'41) we have the first subject presented in thirds with the motto theme in the bass. He will do this twice more before the fugue is finished: in bar 103 (4'28—a particularly beautiful combination), and for the final entrance of the subjects in bar 115 (5'00). I take advantage of the possibility of doubling the motto theme in octaves for the last entry in this extraordinary piece. Note, too, the touching drop from F sharp to F natural in bar 98 (4'15), where the soprano must breathe while the alto holds the tied note.

Contrapunctus 11: When I began work on Contrapunctus 11, I was writing to a distinguished colleague on another matter, and mentioned the hours and hours I was putting into learning this piece. His answer: ‘Ah, our collective nemesis, and one of Bach’s most terrifying pieces!’ Another colleague, an equally distinguished Bach scholar, told me he simply gave up when he got to this point. Even Bach had a crisis with No 11. His plan was to take the three subjects of No 8 and write another fugue with them upside down. That’s exactly what he did, except he realized that it doesn’t quite work. In order to do so, he had to slightly modify the second subject of No 8 (in No 11 presented as the third subject) to make it feasible.

I think it’s important to give this fugue a different tempo from No 10. It is somehow more objective and shouldn’t linger. The first section of the contrapunctus uses the motto theme five times, with much use of its tail in the accompanying parts. The next section opens in bar 27 (1'00) with the entrance of the second subject—quite different in character in its inversion (this was one of Bach’s problems—it simply doesn’t resolve when it goes upwards). But at the same time comes a stroke of genius: Bach chose to juxtapose this second subject with a chromatic scale that from then on appears just about non-stop through the rest of the fugue, both ascending and descending. In that way it is similar to the great Ricercare of the Musical Offering (now known to have been written after this contrapunctus). At bars 57 (2'05) and 67 (2'29), Bach turns the second subject the ‘right’ way up again, bringing this section to a close.

The third section (2'37) has four entries of the motto theme, again turned the right way up as it was in No 8, and closing in F major. At the end of bar 89 (3'18), the third and final subject (that repeated-note one) is introduced, along with the second subject. Now the going gets complicated. For the next 57 bars the insistent repetitions of this subject become more emphatic, and, thanks also to the chromaticisms, the music pushes more and more at the limits of tonality. So it comes as something of a relief when we fall into C major at the beginning of bar 146 (5'20). This is the point (and within one bar of exactly the same point in No 8) where Bach combines the three subjects together for the first time. The most emotionally intense part of the fugue comes at bar 158 (5'45) where, as in the coda of Contrapunctus 5, he gives us the motto theme both the right way up and upside down simultaneously. He does that once more in the left hand in bar 164 (5'58), and then gives us two more combinations of the three subjects before resolving this rather tortuous piece on a major chord.

Contrapunctus 12 (Rectus and Inversus): Having got this far in his master plan, Bach wasn’t going to give up. Not satisfied with inverting only subjects, he decided to write two contrapuncti which could be completely turned upside down, from the first note to the last. These ‘mirror’ fugues couldn’t be more contrasting.

Contrapunctus 12, a four-voice fugue in triple time, is very archaic-sounding. David Schulenberg, in his book The Keyboard Music of J.S. Bach, points out the similarity between this piece and a ‘mirror’ chorale setting by Buxtehude (Mit Fried und Freud) written in 1674. Much of the Rectus version is in the lower part of the keyboard, giving it a very dark quality. A slow tempo, resembling that of a sarabande, seems advisable and gives you the time needed for the unusually big stretches. Beginning in bar 21 (1'03), Bach fills in the intervals in the subject, giving it a greater sense of flow. There are some hints at stretto that don’t materialize. The pedal point in bar 50 (2'34) supports the last entrance of the subject in the tenor, with false entries in the upper voices. On the piano, played by only two hands, this piece needs a great deal of clarity if it is not to sound muddy. The Inversus is more filled with light, and the cadential flourish at the end, which in the ascending Rectus version is a gesture of hope, becomes one of finality when descending in the bass.

Contrapunctus 12 (Rectus and Inversus): Having got this far in his master plan, Bach wasn’t going to give up. Not satisfied with inverting only subjects, he decided to write two contrapuncti which could be completely turned upside down, from the first note to the last. These ‘mirror’ fugues couldn’t be more contrasting.

Contrapunctus 12, a four-voice fugue in triple time, is very archaic-sounding. David Schulenberg, in his book The Keyboard Music of J.S. Bach, points out the similarity between this piece and a ‘mirror’ chorale setting by Buxtehude (Mit Fried und Freud) written in 1674. Much of the Rectus version is in the lower part of the keyboard, giving it a very dark quality. A slow tempo, resembling that of a sarabande, seems advisable and gives you the time needed for the unusually big stretches. Beginning in bar 21 (1'03), Bach fills in the intervals in the subject, giving it a greater sense of flow. There are some hints at stretto that don’t materialize. The pedal point in bar 50 (2'34) supports the last entrance of the subject in the tenor, with false entries in the upper voices. On the piano, played by only two hands, this piece needs a great deal of clarity if it is not to sound muddy. The Inversus is more filled with light, and the cadential flourish at the end, which in the ascending Rectus version is a gesture of hope, becomes one of finality when descending in the bass.

Contrapunctus 13 (Rectus and Inversus): Bach must have realized that he was demanding a lot in asking for No 12 to be played by only ten fingers. Perhaps had he lived longer he might have done what he did for Contrapunctus 13: he wrote a second version of it for two keyboards, four hands, adding a fourth voice to the already existing three. It is, however, not impossible for only one person to play it. It just needs a lot of lightness and agility, a few broken leaps, and, on the piano, some judicious use of the pedal. The variant of the motto theme is a dance-like, playful one in triplets—much needed after the sombreness of the previous contrapunctus (and a theme that inspired Brahms in the last movement of his Cello Sonata No 1 in E minor). Not only does Bach turn this fugue upside down, he also turns it inside out: the top becomes the middle; the middle becomes the bass; and the bass becomes the top!

Contrapunctus 13 (Rectus and Inversus): Bach must have realized that he was demanding a lot in asking for No 12 to be played by only ten fingers. Perhaps had he lived longer he might have done what he did for Contrapunctus 13: he wrote a second version of it for two keyboards, four hands, adding a fourth voice to the already existing three. It is, however, not impossible for only one person to play it. It just needs a lot of lightness and agility, a few broken leaps, and, on the piano, some judicious use of the pedal. The variant of the motto theme is a dance-like, playful one in triplets—much needed after the sombreness of the previous contrapunctus (and a theme that inspired Brahms in the last movement of his Cello Sonata No 1 in E minor). Not only does Bach turn this fugue upside down, he also turns it inside out: the top becomes the middle; the middle becomes the bass; and the bass becomes the top!

Canon per augmentationem in contrario motu: Already with Contrapunctus 12 there is some discussion of which version is the Rectus and which is the Inversus (I choose the Rectus to be the one that starts with the subject in its inversion). But that debate is nothing compared to the problems we face at this point regarding what should come next. In the first edition the four canons are placed before what is now known as Contrapunctus 14, and in the order in which I present them on this recording. Many scholars think that the first one, the augmentation canon, should be placed last in the group. Perhaps that is true. But I can tell you this: when you are performing The Art of Fugue complete, it’s a very good idea to separate Contrapunctus 13 and the canon at the octave, unlike the order they advocate. Otherwise you have two lively pieces, both using triplets, next to each other. By placing the augmentation canon at the beginning of the group, we are afforded more contrast.

When you listen to this piece of music without looking at the score, you might wonder what on earth you are listening to. Looking at it on the page, you immediately see what Bach has done. A very beautiful variant of the motto theme, gently paced, starts off this canon in the top voice. Then the lower voice enters with the same music turned upside down and its note values stretched to twice their length (0'10). Halfway through he reverses the leadership role, giving the leading voice to the bass (2'06).

Canon alla ottava: It is good to have a brilliant movement at this point and the canon at the octave fulfils that role well. The jig-like motto theme variant in 9/16 time is inverted at bar 41 (0'50). This is a canon in perpetuity, meaning that it could go on for ever without finishing (emphasized by the repeat sign which I have chosen not to observe). Bach does, however, bring it to a close by adding a coda of five bars (2'04) in which the right hand dramatically crosses over the left.

Canon alla decima in contrapunto alla terza: The next canon has a double time signature, both 4/4 and 12/8. It begins in sombre mood with the inverted subject in the bass—now syncopated and with the tail dotted. When the right hand then enters, at the interval of a tenth (0'16), the left hand introduces triplets that provide a gentle accompaniment to the subject. These triplets are soon shortened to become sextuplets. They must stay light and unhurried if the piece is to maintain its gentle character. As in the augmentation canon, the parts are reversed halfway through at bar 40 (2'10). I choose not to assimilate the dotted rhythm with the triplet, so as to emphasize the dual time signature. To bring this canon to a conclusion, Bach diminishes the syncopated subject in bar 79 (4'19) in the left hand with two bars that are definitely in 4/4 and not 12/8. The music comes to a pause, at which point he writes the word ‘cadenza’ (4'27). I add a simple flourish in the right hand, giving the triplets a last chance to appear.

Canon alla duodecima in contrapunto alla quinta: This last canon uses a very attractive variant of the motto theme as its subject. It sounds rather like a two-part invention or one of the Vier Duette from Clavierübung III. It is another perpetual canon in which the leadership role reverses at bar 34 (0'54).

The canons of The Art of Fugue, unlike those in the 'Goldberg' Variations, have no third voice to accompany the dialogue. But there is beauty to be found in severity, and we should let them speak simply without trying to add too much in the way of ‘interpretation’.

Contrapunctus 14 (Fuga a 3 soggetti): Now we come to the great enigma: Contrapunctus 14, or ‘Fuga a 3 soggetti’ (‘Fugue with three subjects’) as it was called in the first edition; the fugue that Bach left incomplete upon his death. Some scholars think that it doesn’t even belong in The Art of Fugue. For me it does. I only have to play the opening six bars, after all that has come before, and I feel this is its rightful place. The first subject is drastically simple, hugely expressive despite its bleakness. For 114 bars it is used, also in stretto and inversion, to give us some of the most beautiful and perfect part-writing in all the keyboard works of Bach. In bar 114 (4'40), in the alto voice, the second subject enters—more flowing to stand out easily against the first subject. In bar 148 (5'52) these two subjects are combined for the first time. I find the combination in bar 169 (6'37), in F major, particularly beautiful. The climax of this section comes when, beginning in bar 180 (6'59), the second subject in the bass supports two entries of the first subject in stretto in the alto and soprano voices.

After a cadence in G minor (7'26), in comes the third subject. This is no less than Bach’s name in musical notation: B in German being B flat and H being B natural. To the four notes are added a cadential gesture and trill. Its inversion in the bass in bar 222 (8'41) produces some especially poignant chromaticisms. The half-close in bar 233 (9'07) leads us to the point where Bach combines these three subjects together: the first one in the bass, the second in the alto, and Bach’s signature between them in the tenor. Then the music stops. At this point in the first edition, C P E Bach inserted the words: ‘NB. While working on this fugue, in which the name BACH appears in the counter-subject, the author died.’

But what of the main subject of The Art of Fugue? It’s not there. It wasn’t until 1880 that the German pianist, composer and musicologist (and friend of Brahms) Gustav Nottebohm discovered that these three subjects could be combined with the motto theme, thus producing a quadruple fugue. C P E Bach had already hinted at this in the obituary he wrote with Agricola (Bach’s student and son-in-law), published in 1754, in which he said of The Art of Fugue: ‘This is the last work of the author, which contains all sorts of counterpoints and canons, on a single principal subject. His last illness prevented him from completing this project, bringing the next-to-the-last fugue to completion and working out the last one, which was to contain four themes that would then have been inverted note for note in all four voices.’

So was there to have been yet another fugue after this one? Perhaps so. The eminent scholar Christoph Wolff believes that Bach finished this quadruple fugue, but that the end has been lost. Another possibility is that he simply was too ill to finish it. But what if he purposely chose to leave it unfinished? Laurence Dreyfus in his book Bach and the Patterns of Invention argues convincingly for that theory. He writes: ‘The result is that we, the fortunate inheritors of this work, are compelled to wonder at the conclusion to the final fugue at the same time that we reflect on the loss of someone so passionately dedicated to its cultivation.’

Several people have written conclusions to Contrapunctus 14, including Tovey. I have chosen to leave it incomplete. The effect created, especially in performance, when such eternally flowing music comes to a stop takes our breath away and creates an even greater sense of awe than if it were finished.